Is ${506025}$ divisible by $9$ ?
A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {506025}= &&{5}\cdot100000+ \\&&{0}\cdot10000+ \\&&{6}\cdot1000+ \\&&{0}\cdot100+ \\&&{2}\cdot10+ \\&&{5}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {506025}= &&{5}(99999+1)+ \\&&{0}(9999+1)+ \\&&{6}(999+1)+ \\&&{0}(99+1)+ \\&&{2}(9+1)+ \\&&{5} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {506025}= &&\gray{5\cdot99999}+ \\&&\gray{0\cdot9999}+ \\&&\gray{6\cdot999}+ \\&&\gray{0\cdot99}+ \\&&\gray{2\cdot9}+ \\&& {5}+{0}+{6}+{0}+{2}+{5} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${506025}$ is divisible by $9$ if ${ 5}+{0}+{6}+{0}+{2}+{5}$ is divisible by $9$ Add the digits of ${506025}$ $ {5}+{0}+{6}+{0}+{2}+{5} = {18} $ If ${18}$ is divisible by $9$ , then ${506025}$ must also be divisible by $9$ ${18}$ is divisible by $9$, therefore ${506025}$ must also be divisible by $9$.